Theorem isocnv 5492

Description: Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by set.mm contributors, 27-Apr-2004.)

Assertion

Ref	Expression
isocnv	⊢ (H Isom R, S (A, B) → ◡H Isom S, R (B, A))

Proof of Theorem isocnv

Dummy variables x y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1ocnv 5300	. . . 4 ⊢ (H:A–1-1-onto→B → ◡H:B–1-1-onto→A)
2	1	adantr 451	. . 3 ⊢ ((H:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y))) → ◡H:B–1-1-onto→A)
3		f1ocnvfv2 5478	. . . . . . . 8 ⊢ ((H:A–1-1-onto→B ∧ z ∈ B) → (H ‘(◡H ‘z)) = z)
4	3	adantrr 697	. . . . . . 7 ⊢ ((H:A–1-1-onto→B ∧ (z ∈ B ∧ w ∈ B)) → (H ‘(◡H ‘z)) = z)
5		f1ocnvfv2 5478	. . . . . . . 8 ⊢ ((H:A–1-1-onto→B ∧ w ∈ B) → (H ‘(◡H ‘w)) = w)
6	5	adantrl 696	. . . . . . 7 ⊢ ((H:A–1-1-onto→B ∧ (z ∈ B ∧ w ∈ B)) → (H ‘(◡H ‘w)) = w)
7	4, 6	breq12d 4653	. . . . . 6 ⊢ ((H:A–1-1-onto→B ∧ (z ∈ B ∧ w ∈ B)) → ((H ‘(◡H ‘z))S(H ‘(◡H ‘w)) ↔ zSw))
8	7	adantlr 695	. . . . 5 ⊢ (((H:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y))) ∧ (z ∈ B ∧ w ∈ B)) → ((H ‘(◡H ‘z))S(H ‘(◡H ‘w)) ↔ zSw))
9		f1of 5288	. . . . . . 7 ⊢ (◡H:B–1-1-onto→A → ◡H:B–→A)
10	1, 9	syl 15	. . . . . 6 ⊢ (H:A–1-1-onto→B → ◡H:B–→A)
11		ffvelrn 5416	. . . . . . . . 9 ⊢ ((◡H:B–→A ∧ z ∈ B) → (◡H ‘z) ∈ A)
12		ffvelrn 5416	. . . . . . . . 9 ⊢ ((◡H:B–→A ∧ w ∈ B) → (◡H ‘w) ∈ A)
13	11, 12	anim12dan 810	. . . . . . . 8 ⊢ ((◡H:B–→A ∧ (z ∈ B ∧ w ∈ B)) → ((◡H ‘z) ∈ A ∧ (◡H ‘w) ∈ A))
14		breq1 4643	. . . . . . . . . . 11 ⊢ (x = (◡H ‘z) → (xRy ↔ (◡H ‘z)Ry))
15		fveq2 5329	. . . . . . . . . . . 12 ⊢ (x = (◡H ‘z) → (H ‘x) = (H ‘(◡H ‘z)))
16	15	breq1d 4650	. . . . . . . . . . 11 ⊢ (x = (◡H ‘z) → ((H ‘x)S(H ‘y) ↔ (H ‘(◡H ‘z))S(H ‘y)))
17	14, 16	bibi12d 312	. . . . . . . . . 10 ⊢ (x = (◡H ‘z) → ((xRy ↔ (H ‘x)S(H ‘y)) ↔ ((◡H ‘z)Ry ↔ (H ‘(◡H ‘z))S(H ‘y))))
18		bicom 191	. . . . . . . . . 10 ⊢ (((◡H ‘z)Ry ↔ (H ‘(◡H ‘z))S(H ‘y)) ↔ ((H ‘(◡H ‘z))S(H ‘y) ↔ (◡H ‘z)Ry))
19	17, 18	syl6bb 252	. . . . . . . . 9 ⊢ (x = (◡H ‘z) → ((xRy ↔ (H ‘x)S(H ‘y)) ↔ ((H ‘(◡H ‘z))S(H ‘y) ↔ (◡H ‘z)Ry)))
20		fveq2 5329	. . . . . . . . . . 11 ⊢ (y = (◡H ‘w) → (H ‘y) = (H ‘(◡H ‘w)))
21	20	breq2d 4652	. . . . . . . . . 10 ⊢ (y = (◡H ‘w) → ((H ‘(◡H ‘z))S(H ‘y) ↔ (H ‘(◡H ‘z))S(H ‘(◡H ‘w))))
22		breq2 4644	. . . . . . . . . 10 ⊢ (y = (◡H ‘w) → ((◡H ‘z)Ry ↔ (◡H ‘z)R(◡H ‘w)))
23	21, 22	bibi12d 312	. . . . . . . . 9 ⊢ (y = (◡H ‘w) → (((H ‘(◡H ‘z))S(H ‘y) ↔ (◡H ‘z)Ry) ↔ ((H ‘(◡H ‘z))S(H ‘(◡H ‘w)) ↔ (◡H ‘z)R(◡H ‘w))))
24	19, 23	rspc2va 2963	. . . . . . . 8 ⊢ ((((◡H ‘z) ∈ A ∧ (◡H ‘w) ∈ A) ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y))) → ((H ‘(◡H ‘z))S(H ‘(◡H ‘w)) ↔ (◡H ‘z)R(◡H ‘w)))
25	13, 24	sylan 457	. . . . . . 7 ⊢ (((◡H:B–→A ∧ (z ∈ B ∧ w ∈ B)) ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y))) → ((H ‘(◡H ‘z))S(H ‘(◡H ‘w)) ↔ (◡H ‘z)R(◡H ‘w)))
26	25	an32s 779	. . . . . 6 ⊢ (((◡H:B–→A ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y))) ∧ (z ∈ B ∧ w ∈ B)) → ((H ‘(◡H ‘z))S(H ‘(◡H ‘w)) ↔ (◡H ‘z)R(◡H ‘w)))
27	10, 26	sylanl1 631	. . . . 5 ⊢ (((H:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y))) ∧ (z ∈ B ∧ w ∈ B)) → ((H ‘(◡H ‘z))S(H ‘(◡H ‘w)) ↔ (◡H ‘z)R(◡H ‘w)))
28	8, 27	bitr3d 246	. . . 4 ⊢ (((H:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y))) ∧ (z ∈ B ∧ w ∈ B)) → (zSw ↔ (◡H ‘z)R(◡H ‘w)))
29	28	ralrimivva 2707	. . 3 ⊢ ((H:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y))) → ∀z ∈ B ∀w ∈ B (zSw ↔ (◡H ‘z)R(◡H ‘w)))
30	2, 29	jca 518	. 2 ⊢ ((H:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y))) → (◡H:B–1-1-onto→A ∧ ∀z ∈ B ∀w ∈ B (zSw ↔ (◡H ‘z)R(◡H ‘w))))
31		df-iso 4797	. 2 ⊢ (H Isom R, S (A, B) ↔ (H:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y))))
32		df-iso 4797	. 2 ⊢ (◡H Isom S, R (B, A) ↔ (◡H:B–1-1-onto→A ∧ ∀z ∈ B ∀w ∈ B (zSw ↔ (◡H ‘z)R(◡H ‘w))))
33	30, 31, 32	3imtr4i 257	1 ⊢ (H Isom R, S (A, B) → ◡H Isom S, R (B, A))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2615   class class class wbr 4640  ^◡ccnv 4772  –→wf 4778  –1-1-onto→wf1o 4781   ‘cfv 4782   Isom wiso 4783

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-iso 4797

This theorem is used by:  isores1  5495